Symmetry and magnitude of spin–orbit torques in
ferromagnetic heterostructures
Kevin Garello
1
*
, Ioan Mihai Miron
2
, Can Onur Avci
1
, Frank Freimuth
3
, Yuriy Mokrousov
3
,
Stefan Blu
¨
gel
3
,Ste
´
phane Auffret
2
, Olivier Boulle
2
, Gilles Gaudin
2
and Pietro Gambardella
1,4,5
*
Recent demonstrations of magnetization switching induced by in-plane current injection in heavy metal/ferromagnetic
heterostructures have drawn increasing attention to spin torques based on orbital-to-spin momentum transfer. The
symmetry, magnitude and origin of spin–orbit torques (SOTs), however, remain a matter of debate. Here we report on the
three-dimensional vector measurement of SOTs in AlO
x
/Co/Pt and MgO/CoFeB/Ta trilayers using harmonic analysis of
the anomalous and planar Hall effects. We provide a general scheme to measure the amplitude and direction of SOTs as a
function of the magnetization direction. Based on space and time inversion symmetry arguments, we demonstrate that
heavy metal/ferromagnetic layers allow for two different SOTs having odd and even behaviour with respect to
magnetization reversal. Such torques include strongly anisotropic field-like and spin transfer-like components, which
depend on the type of heavy metal layer and annealing treatment. These results call for SOT models that go beyond the
spin Hall and Rashba effects investigated thus far.
M
emory and logic spintronic devices rely on the generation
of spin torques to control the magnetization of nanoscale
elements using electric currents
1,2
. Conventionally, such
torques have been associated with the transfer of spin angular
momentum between a ‘polarizer’ and a ‘free’ ferromagnetic layer
separated by a non-magnetic spacer, mediated by a spin-polarized
current flowing perpendicular to the two layers
2,3
. Recently,
however, experiments
4–13
and theory
14–27
have pointed out alter-
native mechanisms to produce spin torques that do not require a
polarizer ferromagnetic layer. These mechanisms, which include
the spin Hall
28
, Rashba
29
and Dresselhaus
30
effects, exploit the
coupling between electron spin and orbital motion to induce
non-equilibrium spin accumulation, which eventually gives rise to
a torque on the magnetization via the spin transfer between s and
d electrons
31,32
. Henceforth, we refer to such phenomena as
spin–orbit torques (SOTs) to underline their common link to the
spin–orbit interaction.
Of particular relevance for magnetization switching, experiments
on AlO
x
/Co/Pt heterostructures have shown that current injection
in the plane of the layers induces a spin accumulation component
transverse to the current, dm
⊥
≈ z × j (refs 5,6), as well as a longi-
tudinal one that rotates with the magnetization in the plane defined
by the current and the z-axis of the stack, dm
≈ (z × j) × m (refs
9,33), where j and m are unit vectors that denote the current
density and magnetization direction, respectively. Because of the
exchange interaction between s and d electrons, these components
produce two effective magnetic fields, B
⊥
≈ dm
⊥
and B
≈ dm
,
or, equivalently, a field-like torque T
⊥
≈ m × dm
⊥
and a spin trans-
fer-like torque T
≈ m × dm
.Ifj is injected along x, these torques
correspond to T
⊥
≈ m × y and T
≈ m × (y × m), respectively.
Several studies have shown that T
is strong enough to reverse the
magnetization of high-coercivity ferromagnetic layers with both
perpendicular
9,33,34
and in-plane
35
anisotropy for current densities
of the order of 10
7
–10
8
Acm
22
, raising interest in SOTs for
technological applications. For example, it has been proposed
9,36,37
and demonstrated
35
that T
can be used to induce switching of mag-
netic tunnel junction devices using a three-terminal configuration,
where the read and write current paths are separated to avoid
damage to the tunnel barrier.
On the theoretical side, two apparently contrasting pictures have
emerged: one based on the bulk spin Hall effect (SHE) in the heavy
metal layer as the sole source of spin accumulation
9,20,23–25,33,35
and
the other on Rashba-type effective fields and spin-dependent scatter-
ing, which take place at the interface between the heavy metal and
the ferromagnetic layer
9,21–26
. Both pictures lead to qualitatively
equivalent expressions for T
⊥
and T
(refs 23–25) but differ in
the relative magnitude of the torques, because a pure SHE implies
T
≫ T
⊥
, whereas the opposite is expected if only interfacial
Rashba fields are considered
25
. Experiments by Liu et al. have
shown that the SHE dominates the contribution to T
and that its
sign is reversed in MgO/CoFeB/Ta and AlO
x
/Co/Pt, consistently
with the opposite sign of the SHE in Ta and Pt
33,35
. Recent data,
however, show that the magnitude and even the sign of both T
and T
⊥
in MgO/CoFeB/Ta depend on the thickness of the Ta
layer
38
, suggesting that different effects contribute to these torques.
This state of affairs, together with the lack of consistent methods
to measure the torques, makes it hard to optimize the SOT efficacy
for applications and reach a consensus on their physical origin.
The purpose of this Article is threefold. First, starting from sym-
metry arguments, we derive general expressions of the spin accumu-
lation and current-induced SOTs in magnetic heterostructures that
are independent of specific physical models. Second, we present a
self-consistent, sensitive method to perform three-dimensional
vector measurements of SOTs using an a.c. susceptibility technique
based on the combination of the 1st and 2nd harmonic contri-
butions of the anomalous Hall (AHE) and planar Hall (PHE)
effects. Third, we demonstrate unambiguously the existence of
two distinct SOTs that have odd and even symmetry with respect
1
Catalan Institute of Nanoscience and Nanotechnology (ICN2), E-08193 Barcelona, Spain,
2
SPINTEC, UMR-8191, CEA/CNRS/UJF/GINP, INAC, F-38054
Grenoble, Fr ance,
3
Peter Gru¨nberg Institut and Institute for Advanced Simulation, Forschungszentrum Ju¨lich and JARA, 52425 Ju¨lich, Germany,
4
Institucio
´
Catalana de Recerca i Estudis Avanc¸ats (ICREA), E-08010 Barcelona, Spain,
5
Department of Materials, ETH Zurich, CH-8093 Zurich, Switzerland.
*
e-mail: kevin.garello@mat.ethz.ch; pietro.gambardella@mat.ethz.ch
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to the inversion of the magnetization and include, but are not
limited to, T
⊥
≈ m × y and T
≈ m × (y × m) (Fig. 1a–c). We
find strongly anisotropic SOT components that have not been
observed to date, which depend on the x and y projections of the
magnetization in the plane of the current. T
⊥
and T
have compar-
able magnitude in AlO
x
/Co/Pt and decrease significantly due to
interface diffusion upon annealing. Both T
⊥
and T
reverse sign
and are dominated by anisotropy effects in MgO/CoFeB/Ta. The
picture that emerges from this study is that interfacial effects play
a prominent role in determining the magnitude and anisotropy of
the torques.
Spin–orbit torque symmetry and effective fields
SOTs require inversion asymmetry in order to induce net effects on
the magnetization, which is usually realized by sandwiching a ferro-
magnetic layer between two dissimilar layers (Fig. 1). This holds also
for torques produced by the SHE, which average to zero in sym-
metric heterostructures. In Supplementary Section S1 we derive
the general expressions for dm
⊥,
and SOTs consistent with the
minimal requirements imposed by structure inversion asymmetry,
namely rotational invariance around the z-axis and mirror sym-
metry with respect to planes parallel to z. We find that the spin
accumulation contains magnetization-dependent terms that add
to the dm
⊥
≈ y and dm
≈ y × m components considered thus
far, which change the symmetry and amplitude of T
⊥
and T
. The
quantitative significance of these terms, however, must be estab-
lished by experiment. Our measurements determine a minimal set
of terms required to model the action of the field-like and spin
transfer-like torques, namely
T
⊥
=(y × m) T
⊥
0
+ T
⊥
2
z × m()
2
+ T
⊥
4
z × m()
4
+ m × z × m()m · x()T
⊥
2
+ T
⊥
4
z × m()
2
(1)
T
= m ×(y × m) T
0
+ z × m()m · x()T
2
+ T
4
z × m()
2
(2)
These torques are, respectively, odd and even with respect to the
inversion of m. For the special case T
⊥
n
= T
n
= 0 for all n = 0,
equations (1) and (2) simplify to T
⊥
= T
⊥
0
(y × m) and
T
= T
0
m ×(y × m), which have been obtained theoretically for
several models discussed before
21–26
.
For the purpose of comparison with the experiment, we consider
here the effective magnetic fields B
⊥
and B
perpendicular to the
magnetization that correspond to T
⊥
and T
obtained above. We
adopt a spherical coordinate system (Fig. 1d), where m ¼ (sin
u
cos
w
, sin
u
sin
w
, cos
u
) and obtain
B
⊥
=−cos
u
sin
w
T
⊥
0
+ T
⊥
2
sin
2
u
+ T
⊥
4
sin
4
u
e
u
− cos
w
T
⊥
0
e
w
(3)
B
= cos
w
T
0
+ T
2
sin
2
u
+ T
4
sin
4
u
e
u
− cos
u
sin
w
T
0
e
w
(4)
Using equations (3) and (4), the action of the current-induced fields
on the magnetization can be directly compared to that of a reference
external field (B
ext
) of known magnitude and direction by means of
low-frequency susceptibility measurements
6,39
.
Hall measurements of current-induced effective fields
We studied AlO
x
(2 nm)/Co(0.6 nm)/Pt(3 nm) as a model system,
patterned into 1 × 1 and 1 × 0.5 mm
2
rectangular dots (Fig. 1).
We used an a.c. current of frequency f to modulate the SOT ampli-
tude and induce small oscillations of m about its equilibrium direc-
tion, defined by B
ext
and the magnetic anisotropy of the trilayer.
Such oscillations generate a second-harmonic contribution to the
Hall voltage (V
H
), which provides a sensitive way to measure
current-induced fields (Supplementary Section S2). In general, V
H
depends on m
z
through the AHE and on the product m
x
m
y
through the PHE:
V
H
= R
AHE
I cos
u
+ R
PHE
I sin
2
u
sin 2
w
(5)
−0.06
−0.03
0.00
−0.4
0.4
a
b
c
d
e
f
R
f
H
(Ω)
B
ext
(T) B
ext
(T)
0.060.03
0.0
−0.4
0.4
R
f
H
(Ω)
0.0
−1.0 −0.5 0.0 1.00.5
φ
θ
B
θ
B
= 0° θ
B
= 82°
x
y
z
e
φ
e
θ
m
B
ext
I
II I
T
T
T
T
T
T
m
θ
m
m
Figure 1 | Torque schematics and magnetization measurements. a–c,AlO
x
/Co/Pt Hall cross with current and voltage leads. The thick arrows indicate the
direction and amplitude of T
⊥
(red) and T
(blue) for
w
¼ 08 (a),
w
¼ 608 (b)and
w
¼ 908(c). The thin arrows indicate the equivalent fields B
⊥
(red) and B
(blue). d, Coordinate system. e,f, m
z
measured by the first harmonic Hall resistance R
f
H
as a function of B
ext
applied parallel to the easy axis (
u
B
¼ 08)(e)and
close to in-plane (
u
B
¼ 828,
w
¼ 08)(f).
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where R
AHE
and R
PHE
are the AHE and PHE resistances, respect-
ively, and I is the injected current. In terms of the total Hall resist-
ance R
H
¼ V
H
/I, the first harmonic term R
f
H
= R
f
AHE
+ R
f
PHE
relates
to the equilibrium direction of the magnetization and is indepen-
dent of modulated fields. The second harmonic term R
2f
H
measures
the susceptibility of the magnetization to the current-induced fields
and is given by
R
2f
H
= R
AHE
− 2R
PHE
cos
u
sin 2
w
d cos
u
dB
ext
B
u
sin(
u
B
−
u
)
+ 2R
PHE
sin
2
u
cos 2
w
B
w
B
ext
sin
u
B
(6)
where B
u
and B
w
represent the polar and azimuthal components of
the total effective field B
⊥
þ B
induced by the current,
u
B
is the
polar angle of B
ext
, and
w
;
w
B
(Fig. 1d). Equation (6) allows us
to measure B
u
and B
w
as a function of
u
and
w
.IfR
PHE
¼ 0, it is
straightforward to evaluate B
u
by noting that
R
AHE
d cos
u
dB
ext
=
dR
f
H
dB
ext
Otherwise, B
u
and B
w
must be evaluated by measuring V
H
at
w
¼ 08
and 908 and fitting R
2f
H
using a recursive procedure that
accounts for both the AHE and the PHE (R
AHE
¼ 0.72 V and
R
PHE
¼ 0.09 V for the sample presented in Figs 1– 4). This
method has been validated by numerical macrospin simulations
as well as by applying external a.c. fields in phase and antiphase
with the current, the amplitude of which was recovered using
equation (6) (Supplementary Sections S4 and S6).
Figure 1e,f shows R
f
H
as a function of B
ext
applied out of plane
(
u
B
¼ 08) and nearly in-plane (
u
B
¼ 828), respectively. The curves,
proportional to m
z
, are characteristic of AlO
x
/Co/Pt layers with
strong perpendicular magnetic anisotropy. The slow and reversible
10
15
20
−6
−3
0
3
6
0.0
1.5
3.0
a
b
c
d
e
f
−1.0 −0.5 0.0
B
ext
(T)
1.00.5
−1.0 −0.5 0.0
B
ext
(T)
1.00.5
−1.0 −0.5 0
B
ext
(T)
1.00.5
−1.0 −0.5 0.0
B
ext
(T)
1.00.5
R
H
2f
(mΩ)
B
/cos
θ
(mT)
B
/cos
θ
(mT)
B
||
(mT)
B
||
(mT)
R
H
2f
(mΩ)
φ = 90°
φ = 0°
−20
0
20
0.0 0.2 0.4 0.6
10
15
20
sin
2
θ
θ (deg)
20 30 40 50
18
19
20
21
Figure 2 | Second-harmonic Hall resistance and current-induced spin–orbit fields. a,b, R
2f
H
measur ed as a function of B
ext
applied at
u
B
¼ 828 and
w
¼ 908 (a),
and
u
B
¼ 828 and
w
¼ 08 (b). The amplitude of the a.c. current is 1.136 mA. c, Effectiv e field B
⊥
/cos
u
measured at
w
¼ 908 as a function of B
ext
. d, Effective
field B
measured at
w
¼ 08 as a function of B
ext
. e, B
⊥
/cos
u
measured at
w
¼ 908 as a function of sin
2
u
. The solid line is a fit to T
⊥
0
+ T
⊥
2
sin
2
u
according
to equation (3). f, B
measured at
w
¼ 08 as a function of
u
. The solid line is a fit to T
0
+ T
2
sin
2
u
+ T
4
sin
4
u
according to equation (4). Note that
T
⊥
= B
⊥
/ cos
u
for
w
¼ 908 and T
= B
for
w
¼ 08.
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decrease of R
f
H
with increasing in-plane field observed in Fig. 1f is
due to the coherent rotation of the Co magnetization towards the
hard plane direction. Figure 2 shows the second-harmonic measure-
ments of R
2f
H
as a function of B
ext
applied at
u
B
¼ 828, perpendicular
(
w
¼ 908, Fig. 2a) and parallel (
w
¼ 08, Fig. 2b) to the current. The
data are shown after subtraction of sample-dependent contributions
to the Hall voltage that are not included in equation (5), namely a
constant offset due to the voltage probe asymmetry as well as the
anomalous Nernst–Ettinghausen effect (ANE). The ANE can be
separately measured, giving a small correction to R
2f
H
of the order
of 0.1 mV (Supplementary Section S7). We note that the choice
of
u
B
is not critical as long as B
ext
is slightly tilted off-plane, to
prevent the formation of magnetic domains. According to
equation (6), R
2f
H
is mostly sensitive to the effective field components
parallel to e
u
, as these affect m
z
and hence the AHE. Conversely, the
components parallel to e
w
are measured through the PHE, which is
significantly weaker. Thus, R
2f
H
measured at
w
¼ 908 reflects
mostly B
⊥
contributions, whereas R
2f
H
measured at
w
¼ 08 reflects
mostly B
terms. This agrees with the even/odd character of R
2f
H
measured at
w
¼ 908/08 with respect to field inversion (Fig. 2a,b),
because B
⊥
and B
are even and odd with respect to m, opposite
to the torques from which they are derived.
Field-like and spin transfer-like torque components
The effective fields B
⊥
and B
derived from R
2f
H
for m // y (
w
¼ 908)
and m // x (
w
¼ 08), respectively, are shown in Fig. 2c,d. We find
several interesting features that reveal a more complex scenario
than previously anticipated. In particular, B
⊥
depends strongly on
the direction of m, which is determined here by B
ext
. By converting
the field dependence into a
u
dependence using the AHE, we find
that B
⊥
measured at
w
¼ 908 closely follows the function
−cos
u
(T
⊥
0
+ T
⊥
2
sin
2
u
), with T
⊥
0
=−11. + 0.7 mT and
T
⊥
2
=−11.2 + 0.6 mT (Fig. 2e). This expression agrees with
equation (3), but differs remarkably from that expected from
either the Rashba field
5,14,24–26
or the field-like component of the
SHE torque
23,25
reported in the literature, which imply T
⊥
2
= 0.
We note that T
⊥
0
includes the contribution of the Oersted field
produced by the current flowing in the Pt layer, which we estimate
as
m
0
I/2L =−0.7 mT (antiparallel to y), where L is the width of the
current line and
m
0
the vacuum permeability.
−5
0
5
a
b
c
d
e
f
0
1
2
−1.0 −0.5
−1.0 −0.5
0.0
1.00.5 0.0 1.00.5
−6
−3
0
3
6
0
9
17
27
37
46
57
67
78
90
0
9
17
27
37
46
57
67
78
90
0
9
17
27
37
46
57
67
78
90
φ
(deg)
φ
(deg)
φ
(deg)
φ
(deg)
φ
(deg)
φ
(deg)
10 20 30 40 50 60 70 80 90
5
10
15
10 20 30 40 50 60 70 80 90
5
10
15
15
20
02040608010 30 50 70
R
H
2f
(mΩ)
B
ext
(T)
B
ext
(T)
−1.0 −0.5 0.0 1.00.5
B
ext
(T)
R
H
2f
(B ) (mΩ)
−T
0
(mT)
−T
2
(mT)
T
0
||
(mT)
R
H
2f
(B
||
) (mΩ)
Figure 3 | Angular dependence of the Hall resistance and SOT components. a, R
2f
H
as a function of B
ext
applied at
u
B
¼ 828 measured for different in-plane
orientations of the magnetization. b,c,SymmetricR
2f
H
(B
⊥
) (b) and antisymmetric R
2f
H
(B
) (c) components of R
2f
H
. d–f, SOT components T
⊥
0
(d), T
⊥
2
(e)andT
0
(f) as a function of
w
. The error bars represent the experimental errors, which are mostly due to the uncertainty of the PHE measurements. The amplitude of
the ac current is 1.136 mA.
0
10
20
a
b
c
0.5 × 0.5 μm
2
1 × 1 μm
2
1 × 0.5 μm
2
0.5 × 0.5 μm
2
1 × 1 μm
2
1 × 0.5 μm
2
0.5 × 0.5 μm
2
1 × 1 μm
2
1 × 0.5 μm
2
0
j (10
7
A cm
−2
) j (10
7
A cm
−2
) j (10
7
A cm
−2
)
213
0
5
10
15
0213
0
10
20
30
0213
−T
0
(mT)
−T
2
(mT)
T
0
||
(mT)
Figure 4 | Dependence of the field-like and spin transfer-like SOT components on the injected current density. a–c, T
⊥
0
(a), T
⊥
2
(b)andT
0
(c) as a function
of j for different samples. Red circles and black squares refer to square Hall crosses with 1 × 1
m
m
2
and 0.5 × 0.5
m
m
2
dimensions, respectively . The blue
triangles refer to a narrow Hall cross with a 1-mm-wide current line and 0.5-mm-wide voltage probes.
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The dependence of B
on the magnetization is remarkably
different from B
⊥
. Figure 2f shows that B
measured at
w
¼ 08
is weakly dependent on
u
, and is well-approximated by
T
0
+ T
2
sin
2
u
+ T
4
sin
4
u
, in agreement with equation (4), with
T
0
= 19.0 + 0.5 mT, T
2
= 2 + 1 mT and T
4
=−1 + 1 mT.
As the higher-order coefficients are small and tend to compensate,
B
can be reasonably approximated by a constant value T
0
, consist-
ently with previous findings
33,34
. This behaviour is typical of
as-deposited AlO
x
/Co/Pt samples, apart from small changes of the
coefficients that we attribute to pattern or material inhomogeneities.
To complete the description of B
⊥
and B
, we performed a series of
measurements for different in-plane orientations of m.When
w
devi-
ates from 08 or 908, R
2f
H
, shown in Fig. 3a, is given by the linear super-
position of two terms R
2f
H
(B
⊥
)+R
2f
H
(B
), which can be easily
separated owing to their even/odd symmetry with respect to the
inversion of m.Figure3b,cshowsR
2f
H
(B
⊥
) and R
2f
H
(B
) as a
function of
w
.ThelineshapeofR
2f
H
(B
⊥
) is similar to R
2f
H
measured at
w
¼ 908 (Fig. 2a), whereas R
2f
H
(B
) is similar to R
2f
H
measured
at
w
¼ 08 (Fig. 2b). The amplitude of R
2f
H
(B
⊥
) increases whereas
R
2f
H
(B
) decreases as
w
goes from 08 to 908. Fro m these curves we
obtain that the polar component of B
⊥
scales proportionally to sin
w
,
whereas the polar component of B
scales as cos
w
, in agreem ent
with equations (3) and (4), respectively . This implies that, within the
error of our data, the SOT coefficients T
⊥
0
, T
⊥
2
and T
0
are independent
of
w
(Fig. 3d–f), in agr eement with the superposition principle for the
current and the resulting linear-response torques.
Torque-to-current ratios
Figure 4 shows that the amplitudes of T
⊥
and T
scale linearly with
the current up to j ¼ 1.5 × 10
7
Acm
22
. Above this value, we
observe a nonlinear increase of the coefficients T
⊥
0
, T
⊥
2
and T
0
,
which we attribute to Joule heating. At the maximum current
density used in this study (3.15 × 10
7
Acm
22
), heating induces a
reduction of the AHE (23.5%) and magnetic anisotropy (213%),
as well as an increase in the resistivity of the layers (þ13%). We
caution that these effects can alter the intrinsic SOT/current
ratio and also introduce experimental artefacts (Supplementary
Section S12).
Our measurements also offer quantitative insight into the
magnitude of the different SOT components. We first discuss T
⊥
.
From the initial slope of the data in Fig. 4, we find that the torque/
current ratios corresponding to T
⊥
0
and T
⊥
2
are 23.2+0.2 and
22.3+0.2 mT per 10
7
Acm
22
, respectively. This corrects our pre-
vious estimate of T
⊥
based on current-induced domain nucleation
5
,
which largely overestimated T
⊥
0
due to heat-assisted magnetization
reversal
6
and neglect of T
⊥
2
. Moreover, our measurements are
quasi-static and extend well into the low-current regime, proving
that T
⊥
does not result from the spin Hall torque dynamics at
high current. This hypothesis was suggested by Liu et al., who
reported no evidence of T
⊥
in AlO
x
/Co/Pt within a sensitivity of
1.3 mT per 10
7
Acm
22
(ref. 33). We suggest that the negative
result of Liu et al. might be partly due to the different preparation
of the AlO
x
/Co/Pt stack (oxidized in air and annealed up to
350 8C) and, possibly, to the different measurement method
(see Supplementary Section S10 for a comparison of a.c. and
d.c. measurements).
We next consider T
, fitting the low-current data
( j , 1.5 × 10
7
Acm
22
) in Fig. 4c. We obtain T
0
= 5.0 + 0.2mT
per 10
7
Acm
22
for the square Hall crosses (circles and squares in
Fig. 4c). This represents a lower bound for the torque amplitude
due to current dispersion in the voltage probes, which can reach
up to 23% of the total current
40
. Measurements of Hall crosses
with narrower voltage probes (0.5 mm instead of 1 mm) give consist-
ently higher torque/current ratios, namely T
⊥
0
=−4.0 + 0.3,
30 40
10 20 30 40 50 60
−4.5
−4.0
−3.5
−3.0
10 20 30 40 50 60
−16
−14
−12
−10
−8
−6
20 50
12
16
20
4
5
6
7
10 20 50 60
8
12
16
3
4
5
30 40
As deposited
Annealed
As deposited
Annealed
a
b
c
d
−3.5
−3.0
−2.5
−12
−10
−8
−6
−4
B
/cosθ (mT)
B
/cosθ (mT)
θ (deg)
θ (deg) θ (deg)
θ (deg)
B
||
(mT)
B
||
(mT)
Figure 5 | Effect of thermal annealing and stack composition on current-induced spin–orbit fields. a,b, Effectiv e field B
⊥
/ cos
u
measured at
w
¼ 908 (a)
and effective field B
measured at
w
¼ 08 (b)inAlO
x
/Co/Pt as a function of
u
. The measurements refer to a 0.5 × 0.5
m
m
2
as-deposited sample (black
circles) annealed to 300 8C (orang e triangles). The amplitude of the a.c. current is 540
m
A and 550
m
A for the as-deposited and annealed samples,
respectively. c,d, B
⊥
/cosu measured at
w
¼ 908 (c)andB
measured at
w
¼ 08 (d) for an MgO/CoFeB/Ta Hall bar annealed to 300 8C. The width of the
Hall bar is 1
m
m and the amplitude of the a.c. current is 500
m
A. The scale on the right-hand side of the plots is in mT per 10
7
Acm
22
.
NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2013.145
ARTICLES
NATURE NANOTECHNOLOGY | VOL 8 | AUGUST 2013 | www.nature.com/naturenanotechnology 591
© 2013 Macmillan Publishers Limited. All rights reserved.